Small is Beautiful in Mathematics

In spite of all the advances in technology, the brain is still fairly primitive when it comes to understanding numbers. The brain works better with smaller numbers rather than bigger ones. This is the trick that you use when you need to approximate. Instead of looking at a very big number, you just concentrate on the first one or two digits. So 123 x 321 can be seen as 100 x 300 = 30,000. (A little closer to 40,000 actually). We use this principle in problem solving, by first working with smaller numbers, getting comfortable with the logic, and then applying it to larger numbers.

But the question before us is a broader one. Why should any person, say a student not preparing for any exam, learn words? Other than intangible benefits, such as increased comprehension and understanding, enhanced vocabulary for communication, and supposedly better academic performance, is there any other benefit of learning words? After all, learning words is an inexact science.

Let's get straight to the point:

Here is an example: The pages of Jack's book are numbered from 1. The page numbers have a total of 555 digits. How many pages does the book have?

Let us say that the book had 9 pages, it would have 9 digits. If it had 10 pages it would become 9 + 2 = 11. Now we think bigger: If it had 99 pages, number of digits would be 9 + 90*2 = 189. You get the hang of it. Don't even try with 999 - it would take the answer beyond 555. We can now see that each additional page is adding 3 digits. We need 555 - 189 = 366 digits. So more pages that are required: 366 / 3 = 122. So the answer is 122 + 99 = 221 pages. The answer lies in the relationship between Vocabulary & Success.

Let's take one more example: What is the smallest number with which 20! should be divided so that it becomes odd ?

Now in this case, since you are not aware of the concept, let's take a smaller number, say 40. Try to make it odd. You know a number becomes odd, when it is not divisible by 2. Start dividing it successively by 2, you will see that you can do so only three times and now you have 5 left with you, which is odd. In conceptual terms, a number becomes odd, if all the 2s it is divisible by are removed. Let's come back to the original question: how many times 2 can divide 20! or what is the largest power of 2 in 20! Following the numbers concept, you can find that to be 18. Hence the number is to be divided by 218.

Hope you have got the hang of how to work with small numbers and extrapolate the results to larger numbers. Just to make sure you have learnt something, try your hands at this problem: How many different positive integers exist between 106 and 107, the sum of whose digits is equal to 2?

Best Wishes!!
Team Bulls Eye